Structural Engineering & Statics
where the load goes
Deflection

Stiffer than its cracked section says

At a crack the concrete below the neutral axis has gone and the steel carries the tension alone. Between the cracks it has not gone — bond drags it back into tension, the steel strain drops, and the curvature averaged over a length of beam is neither section's.

Assumes Bending is a pair of forces, pushing and pulling, Stiffness is not strength, and usually it is the one that governs and The deflection that arrives three years late.

20 min read 10 figures Which failure arrives firstThe load that will not hold stillScale changes everything

When half the section has given up is a question about a section: at a crack, the concrete below the neutral axis carries nothing, the steel carries the whole tension, and the second moment of what is left is a third of the gross.

Between the cracks it has not given up. The steel is bonded to the concrete along its whole length, so as soon as the steel tries to stretch it drags the surrounding concrete into tension with it. Halfway between two cracks the concrete has recovered most of the tension it lost, the steel strain has dropped, and the curvature there is much closer to the uncracked value than to the cracked one.

The beam’s deflection is an integral of curvature, so what it responds to is the average.

The beam is stiffer than its cracked section and softer than its gross oneMoment against mid-span deflection for a 300 × 550 mm beam spanning 8.0 m, with the two bounds it lies between. The uncracked line is what the gross transformed section gives; the cracked line is what the section at a crack gives; and the curve between them is the member, because between the cracks the concrete is still carrying tension and the average curvature is not either section's. At the service load the deflection is 23.2 mm — span over 345 — against 8.3 uncracked and 25.2 fully cracked, a factor of 3.03 between the bounds. The interpolation ζ = 1 − β(M_cr/M)² sits it 88 per cent of the way across, and β falls from one to a half under sustained or repeated load because the bond that does the dragging deteriorates.0102030050100150200250mid-span deflection (mm)moment (kNm)uncrackedthe memberfully crackedM_cr = 50 kNm23.2 mm at service, span over 345
Fig. 1 Moment against mid-span deflection, with the two bounds it lies between. The uncracked line is the gross transformed section; the cracked line is the section at a crack; and the curve between them is the member. At the service load the beam deflects 23.2 mm — span over 345 — against 8.3 uncracked and 25.2 fully cracked.

Which free body produced the number

Take a length of beam containing several cracks and ask for the average curvature over it.

At a crack the curvature is M/EcI2M/E_cI_2, the fully cracked value. Midway between cracks it is close to M/EcI1M/E_cI_1, the uncracked one. In between it varies with the bond stress transferring force back into the concrete, and the average over the length is somewhere between the two — closer to the cracked value the higher the moment, because a higher moment means more cracks, closer together, with less uncracked length between them.

The standard interpolation is

1r=ζ(1r)II+(1−ζ)(1r)I,ζ=1−β(McrM)2\frac{1}{r} = \zeta\left(\frac{1}{r}\right)_{II} + (1-\zeta)\left(\frac{1}{r}\right)_{I}, \qquad \zeta = 1 - \beta\left(\frac{M_{cr}}{M}\right)^2

and the thing being interpolated is the curvature, not the second moment. Those are different operations — a weighted average of 1/I11/I_1 and 1/I21/I_2 is not the reciprocal of a weighted average of I1I_1 and I2I_2 — and the difference is largest exactly where the two bounds are furthest apart.

The consequence is worth stating in plain terms: a member’s stiffness is not a property of any of its sections. Every section along the beam is either cracked or not, and neither of them has the stiffness the beam behaves with.

The neutral axis is wherever the first moment vanishesA 300 by 556 section with 1470 mm² of steel at a depth of 500, carrying 176 kNm after the tension side has cracked. The neutral axis is no longer at mid-depth: it has risen to 149.0 mm from the top, which is where the first moment of the compression zone plus the transformed steel vanishes. The compression is 17.5 N/mm² at the top fibre and the steel carries 266 N/mm²; the resulting couple is 391 kN on a lever arm of 450 mm, which multiplies back to the 176 kNm applied. The uncracked section would have had 4664×10⁶ mm⁴ against the cracked 1499×10⁶ — a loss of 68% of the stiffness.x = 1491470 mm² of steel, n = 6.45b = 30017.5 N/mm²391 kN in the steelz = 450C = T = 391 kN · C·z = 176.0 kNm = the applied momentcracked I 1499×10⁶ mm⁴ against uncracked 4664×10⁶ — 68% of the stiffness gone
Fig. 2 The section the lower bound comes from. At a crack this is exactly right: the neutral axis is 149 mm down, the concrete below it is gone, and the steel carries 100 per cent of the tension. The mistake is not in the section analysis; it is in assuming the beam is made of these.

Integrating the curvature, rather than choosing a stiffness

There is a second refinement that matters more than it looks.

Only part of the beam has cracked. Under a uniform load the moment is largest at mid-span and zero at the supports, so a beam whose mid-span moment is 3.5 times the cracking moment is uncracked over the outer fifth of its span and progressively less cracked as the section approaches the ends.

Choosing a single “effective second moment” and using it for the whole beam ignores that. Integrating the curvature — evaluating ζ\zeta at every station and putting the result through the unit-load integral — does not, and the two differ by several per cent on an ordinary beam and by more on a lightly loaded one where the cracked region is short.

This is the same observation about where a deflection comes from, arriving from the other side: the deflection is dominated by the middle half of the span, so what matters most is how cracked the middle half is, and the ends contribute little whatever their state.

Half the beam does nearly all of the deflectingThe virtual-work integrand M·m/EI along the member, normalised to its own peak, with the running share of the answer beside it. The integrand is a density: it says how much of the deflection each millimetre of the beam produced. For this case the middle half supplies 83.7 per cent of it, and the rest of the member supplies the remainder. Stiffening the busy 50 per cent by 1.5 times takes the deflection down by 27.9 per cent; the same material spent on the quiet end takes it down by 5.4 — a factor of 5.2 for the same steel. The map of what is contributing is not the map of where the moment is largest, and the second is the one that gets drawn.00.20.40.60.8100.20.40.60.81along the memberdensity ÷ its peak, and running sharewhere it comes fromrunning totalmiddle half: 83.7% of it · worth 5.2× stiffening the other half
Fig. 3 Which parts of the beam are supplying the answer. The middle half of a uniformly loaded span produces 84 per cent of its own deflection, which is also the part that is cracked — so the fully cracked bound is a better approximation than the geometry suggests, and the uncracked one is worse.

The number that decides it is the one nobody can measure

McrM_{cr} is the tensile strength times a section modulus, and everything above turns on it: it fixes where the curve leaves the uncracked line and it sits squared inside ζ\zeta.

Concrete’s tensile strength has a coefficient of variation around twenty per cent, is not measured directly in any routine test, and is usually inferred from the compressive strength by a correlation. It is comfortably the least reliable number in the calculation.

What that does to the answer depends entirely on where the beam is working.

The least reliable number in the material decides the answer, brieflyWhat a twenty per cent error in the concrete's tensile strength does to a computed deflection, against how far past cracking the beam is. Well past the cracking moment it does almost nothing — at 5.0 times M_cr the spread is 4 per cent — because the section is nearly fully cracked and the interpolation has run out. Just above cracking it does everything: at 1.22 times M_cr the same twenty per cent moves the deflection by 85 per cent. Tensile strength is the property with the widest scatter and the least direct test, and a beam designed to sit near its cracking moment has put the answer on it.123411.21.41.61.8moment ÷ cracking momentdeflection spread for ±20% on f_ctm85% at 1.22 M_crcracking
Fig. 4 What a twenty per cent error in the tensile strength does to a computed deflection, against how far past cracking the beam is. Just above the cracking moment it is worth 80 per cent; at three and a half times it, seven. The beam that is hardest to predict is the one that has only just cracked.

A beam designed to sit near its cracking moment has put its answer on the property with the widest scatter, which is an unusual place for a design to end up and a common one. It happens whenever a member is sized by something other than deflection — by cover, by fire, by a standard depth — and ends up lightly stressed.

The corollary is more useful than the warning. A beam well past cracking is insensitive to the tensile strength, so a heavily loaded member’s deflection is more predictable than a lightly loaded one’s, in the ratio of about ten to one.

Bond deteriorates, and the interpolation knows it

β\beta is the bond factor, and it is 1.0 for a single short-term load and 0.5 for sustained or repeated loading.

The reason is physical. What produces tension stiffening is bond between the bar and the concrete around it, and cycling a load breaks that bond down over a lengthening zone either side of each crack — so less of the concrete is recruited, the average curvature moves towards the fully cracked one, and the beam softens.

On the beam here it is worth about four per cent, from 23.2 mm to 24.2, because the beam is already at ζ=0.92\zeta = 0.92 and there is little tension stiffening left to lose. On a beam near cracking it is worth far more, for exactly the reason the previous section gives.

And it compounds with creep rather than adding to it. Creep multiplies the concrete’s compressive strain and takes the effective modulus down; the loss of bond moves the interpolation towards the cracked bound. The two act on different terms of the same expression, and a long-term deflection calculation that includes only the first is a common and substantial underestimate.

The deflection that arrives years lateThe multiplier on a concrete member's deflection under a sustained load, against time. The elastic deflection arrives on the day the load does and is the 1.0 at the left. After a year it has been multiplied by 3.00, after five years by 3.29, and it approaches 3.38. Nothing has been added to the load and nothing about the strength has changed: this is a serviceability failure arriving on a structure that passed every strength check on the day it was built.1 d10 d100 d2.7 yr27 yr0123time under loaddeflection ÷ the deflection on day one1 year: ×3.005 years: ×3.29the deflection the calculation gives
Fig. 5 The other half of a long-term deflection. Creep works on the modulus in the denominator; loss of bond works on where the member sits between its two bounds; and a beam that has been in service for ten years has had both.

Why the answer sits so close to the cracked bound

The interpolated deflection here is 23.2 mm against bounds of 8.3 and 25.2 — 88 per cent of the way to the cracked one. That is typical, and it is worth understanding why, because it changes what tension stiffening is for.

ζ\zeta contains (Mcr/M)2(M_{cr}/M)^2, and squaring a fraction is unkind to it. At twice the cracking moment ζ\zeta is 0.75; at three times it is 0.89; at the 3.5 times this beam reaches it is 0.92. So a beam at any ordinary service load is nearly fully cracked in the interpolation’s terms, and the uncracked bound — which is a factor of three away — is doing almost nothing.

Tension stiffening is therefore not a way of recovering stiffness on a working beam. It is worth a few per cent there. What it is for is the region just above cracking, where it is worth a factor of two and where the fully cracked calculation is badly wrong — and the region just above cracking is where every lightly loaded member in a building sits.

That inverts the usual presentation. The effect is described as a refinement to a deflection calculation, and it is a refinement on the beams that are easy and the whole answer on the beams that are not.

Which limit arrives firstUtilisation of the strength limit and of the deflection limit, against span. Strength grows as the square of the span and deflection as the fourth power, so the two cross — and past the crossing a beam is sized by how far it moves rather than by what it can carry.0.60.811.21.41.61.8200.511.5span, relative to the firstthey cross heredeflection runs out at 1.35strength runs out at 1.54the limitstrengthdeflection
Fig. 6 Which check is deciding the member. A beam governed by strength is well past its cracking moment and insensitive to everything on this page; a beam governed by deflection is often not, and is exactly where the interpolation is carrying the answer.

Where it shows up as something other than a number

Tension stiffening is usually met as a deflection correction. It has two other appearances that are the same phenomenon wearing different clothes.

Crack width. The width of a crack is the difference between the steel’s extension and the concrete’s over the length between cracks — which is exactly the quantity tension stiffening is about. A calculation of crack width and a calculation of deflection are the same integral with different weights, and a member with more tension stiffening has both a smaller deflection and narrower cracks.

Redistribution. A continuous beam’s moments depend on the relative stiffness of its regions, and the hogging region over a support cracks before the sagging region at mid-span. So the structure softens unevenly, sheds moment from the support to the span, and does it progressively as the load rises — a redistribution nobody chose, produced entirely by the order in which the parts of the member cracked.

The same load, two diagrams, both in equilibriumOne span of a pair of 8 m spans under 22 kN/m, drawn twice. The elastic solution puts 176 kNm over the support and 99 in the span. Reducing the support moment by 30% and taking what statics then gives leaves 123 and 120: the section the beam needs falls from 176 kNm to 123, a saving of 30%. Both curves are in equilibrium with the same load — the mid-span ordinate plus half the support moment is the free moment 176 kNm for either — and the second is legitimate for that reason alone. What it costs is 6.7 milliradians of rotation at the support, which the section has to be able to deliver.012345678-150-100-5050100distance along the span (m)bending moment (kNm, sagging up)elastic 176redistributed 123120 kNmβ = 30% · the section needed falls 176 → 123 kNm · the hinge turns 6.7 mrad
Fig. 7 The redistribution the cracking sequence produces. It is usually discussed as a plastic phenomenon at the ultimate limit state. Most of it happens at service load, for a stiffness reason rather than a strength one, and this essay is that reason.

What a designer actually does with it

The arithmetic above is what a computer does. What a designer does is a span-to-depth ratio, and it is worth saying how the two are related, because the connection is not obvious.

A span-to-depth limit is a deemed-to-satisfy rule: a table of depths, adjusted for the reinforcement ratio and the concrete grade, calibrated so that a member inside it will not exceed span over 250 in the long term. Every quantity in this essay is buried inside that calibration — the interpolation, the bond factor, an assumed creep coefficient, an assumed proportion of permanent load, and an assumed tensile strength.

Which is why the rule is generous where this essay says the answer is uncertain and tight where it says it is not. A lightly reinforced member gets a much smaller allowable ratio than a heavily reinforced one, and that is not because it is weaker; it is because a low reinforcement ratio means a low cracking moment relative to the applied one, which is the sensitive region.

The rule also assumes a beam that has cracked. A member that is genuinely uncracked in service — a prestressed one, or a very lightly loaded one — is three times stiffer than the table allows for, and the table has no way of saying so.

Deflection goes as the fourth power of the spanDeflection against span for a constant load intensity and section, with two slower relationships drawn faintly behind it for comparison: the load itself, which grows in proportion to the span, and the bending moment, which grows as its square. Doubling the span multiplies the deflection by sixteen, while the moment only quadruples.11.522.533.54050100150200250300span, relative to the first1×5×16×81×moment: the squareload: the first powerdeflection: the fourth
Fig. 8 Why depth is what the rule is written in. Deflection goes as the fourth power of the span and the reciprocal of the second moment, so a depth limit is the only single number that can stand in for a deflection calculation at all — and it is standing in for everything on this page.

A slab is the same argument with the numbers moved

Everything above is a beam, and the member that most often has its deflection computed is a slab.

A slab is more sensitive to all of it, for one reason: its reinforcement ratio is far lower. A beam at one per cent steel has a cracked second moment about a third of its gross; a slab at 0.3 per cent has one nearer a fifth, so the two bounds are further apart and the interpolation is carrying more of the answer. And because a slab is shallow, its cracking moment is a larger fraction of the moment it carries — which puts it in exactly the region the sensitivity curve says is worst.

Two consequences follow that a beam does not have. A slab’s deflection calculation is far more sensitive to whether it cracked at all, so a lightly loaded slab that stayed uncracked is several times stiffer than the arithmetic predicts and one that cracked during construction is several times softer. And a slab spanning two ways cracks in one direction before the other, so it is anisotropic in stiffness at service load whatever it was designed to be.

A two-way slab is a one-way slab as soon as it is not squareThe share of the load carried by the strips spanning the short way, against the ratio of the sides. The two families of strips cross at the centre and must deflect equally there, and a strip's deflection goes as the fourth power of its span — so at a ratio of 1.33 the short strips already take 76% and at 2 they take 94%. The panel drawn here is 6000 × 7500 m, a ratio of 1.25, and its short strips take 70.9%. Two-way action is worth having at a ratio of one and worth almost nothing by two.11.522.530.40.50.60.70.80.91long span ÷ short spanshare taken by the short strips6000 × 7500 m: 70.9%by 2 : 1 it is a one-way slab
Fig. 9 The member the calculation is most often asked about, and the one where the two bounds are furthest apart. A two-way slab that has cracked in one direction and not the other is sharing its load in a ratio nobody chose.

Where the model stops

The interpolation is calibrated, not derived. The exponent of two on Mcr/MM_{cr}/M comes from fitting tests; the underlying bond mechanics gives something close to it over the useful range and nothing exact. The expression is a good empirical summary of a real phenomenon rather than a solution to it.

Shrinkage is left out entirely, and it is not small. Restrained shrinkage puts the concrete into tension before any load arrives, which lowers the effective cracking moment — sometimes to half its nominal value — and it curves the member before it is loaded. A great many deflections that are blamed on tension stiffening being overestimated are shrinkage that was left out.

The section is singly reinforced. Compression steel restrains creep and shrinkage substantially, and a beam with a top mat behaves differently in the long term by an amount this calculation has no term for.

And the load is monotonic. The first application of a load cracks the member; the second finds it already cracked and follows a different, softer path. A member’s deflection under a load it has carried before is larger than under the same load applied for the first time, and neither of the curves drawn is that second path.

What the pictures cannot show

The curve in the first figure is smooth. What a real beam does at the cracking moment is jump: a crack forms, that section’s stiffness drops in an instant, and the deflection steps. What is drawn is the average of many such steps at many sections, which is what a beam with fifteen cracks in it does and not what one with two does.

Nor can any figure show the thing a site engineer would ask about, which is whether the beam has cracked. Flexural cracks in a service-load member are a tenth of a millimetre wide and under a finish. The difference between the two bounds on this page is a factor of three in stiffness, and there is no way to tell from outside which of them a particular beam is nearer.

The assumption the figure rests on

Every number here assumes the beam cracked because of the load in the calculation.

That is the assumption most often wrong. Members crack from restrained early thermal contraction while the concrete is a few days old, from restrained drying shrinkage over the following months, from handling if they were precast, and from loads applied during construction that are nowhere in the design case — a stack of blocks, a wet slab above, a propping arrangement that put the member into reverse curvature for a week.

A member that cracked for one of those reasons has an effective McrM_{cr} of nearly zero and behaves as the fully cracked section from the first day. That is the honest upper bound on the deflection of a concrete beam, and the distance between it and the calculation — 25.2 mm against 23.2 here, and 25.2 against 12 on a lightly loaded one — is the width of the answer rather than an error in it.

The crack length at which the strength stops matteringFailure stress against crack length for a toughness of 100 MPa√m, with one steel grade drawn. The falling curve is fracture — Kc divided by Y times the root of pi a — and it does not know what the yield stress is. The horizontal lines are the grades. At 355 N/mm² the two cross at a crack 20.1 mm long. Below that the section yields and the crack is irrelevant; above it the crack decides and the 355 is irrelevant. At a working stress of 2.9 N/mm² the critical crack is 301295.2 mm.501001502000100200300400500crack length, mmstress at failure, N/mm²355 N/mm² crosses at 20.1 mmfracture: the crack decidesat 2.9 N/mm² the critical crack is 301295 mm — off this axis
Fig. 10 Where the tensile strength comes from and what it is worth. It is not a material constant so much as a statistical statement about the largest flaw in the volume being stressed, which is why it scatters, why it falls with member size, and why a calculation that depends on it is a calculation with a range rather than a value.

Where this goes next

x = 1371200 mm² of steel, n = 7.5b = 30018.0 N/mm²371 kN in the steelz = 404C = T = 371 kN · C·z = 150.0 kNm = the applied momentcracked I 1139×10⁶ mm⁴ against uncracked 3422×10⁶ — 67% of the stiffness gone When half the section has given up Bending theory puts the neutral axis through the centroid. That is a consequence, not a rule — and when the tension side cracks, the same reasoning moves the axis somewhere else entirely. 0.60.811.21.41.61.8200.511.5span, relative to the firstthey cross heredeflection runs out at 1.40strength runs out at 1.54the limitstrengthdeflection Stiffness is not strength, and usually it is the one that governs A beam can be nowhere near failure and still be unusable, because it has moved too far. For most long-span members that limit arrives first, and stronger steel does not help at all. 1 d10 d100 d2.7 yr27 yr0123time under loaddeflection ÷ the deflection on day one1 year: ×3.005 years: ×3.29the deflection the calculation gives The deflection that arrives three years late A concrete beam that passes every check on the day it is built goes on deflecting for a decade, and ends up three times where it started. Nothing about the load changed, and nothing about the strength was ever in question. 501001502000100200300400500600crack length, mmstress at failure, N/mm²275 N/mm² crosses at 33.5 mm460 N/mm² crosses at 12.0 mmfracture: the crack decides The flaw that sets the strength A member with a crack twenty millimetres long fails at its yield stress. Make the steel stronger and the crack that does it gets shorter, so the same flaw that was harmless in the weaker grade decides the stronger one.

Named alongside this one

Essays reaching for the same objects. Nobody chose these; they are what the concept index makes visible.

The objects this essay names

Each one links to every other essay that touches it.

BondCracked sectionCracking momentCreepCurvatureDeflectionInterpolationReinforcementRepeated loadSecond moment of areaServiceabilityStiffnessTensile strengthTension stiffeningTransformed section